The construction in Section 4, i.e., exponential map of a discrete \(c\) -concave function, is an optimal transport map and universal in the sense that it can approximate any OT between an absolutely continuous \(\mu \)  and arbitrary \(\nu \) , over a compact Riemannian manifold. It is not, however, a diffeomorphism. As a practical way of optimizing this model to approximate arbitrary \(\nu \)  we suggested smoothing the min operation with soft-min. If this, now a differentiable function, is \(c\) -concave then the smoothed version leads to a diffeomorphism (flow). While we are unable to prove that the soft version is \(c\) -concave, we verified numerically that it indeed leads to a diffeomorphism (see fig:diffeomorphism, Appendix). We leave the question of whether the soft-min operator preserves \(c\) -convexity on the sphere and more general manifolds to future work. Furthermore, it is worth noting that the proposed model can potentially be optimized with other methods than as a flow, for instance by directly optimizing the Wasserstein loss similarly to [1]} in the Euclidean case, or semi-discrete transport methods [2]}. Both would not require the map to be a diffeomorphism. We leave such directions to future work as well.
